# Hierarchical Sliced Wasserstein Distance

**Authors:** Khai Nguyen, Tongzheng Ren, Huy Nguyen, Litu Rout, Tan Nguyen, Nhat Ho  
**arXiv:** [2209.13570](https://arxiv.org/abs/2209.13570)  
**Date:** 2022-09-27  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2209.13570v1/

## Key points

- The Hierarchical Sliced Wasserstein (HSW) distance is a new metric between measures proposed in a recent paper.
- The sliced Wasserstein (SW) distance has been widely used in various application scenarios due to its ability to scale to a large number of supports without suffering from the curse of dimensionality.
- However, estimating the sliced Wasserstein requires a relatively large number of projections in high-dimensional settings, which becomes the main computational bottleneck for applications where the number of supports is relatively small compared with the dimension.
- To address this issue, the authors propose deriving projections by linearly and randomly combining a smaller number of projections called bottleneck projections.
- They explain how these projections are used by introducing Hierarchical Radon Transform (HRT), which is constructed by applying Radon Transform variants recursively.
- The approach is formulated into the new metric between measures named HSW distance.
- By proving the injectivity of HRT, they derive the metricity of HSW and investigate its theoretical properties including its connection to SW variants and its computational and sample complexities.
- Finally, they compare the computational cost and generative quality of HSW with conventional SW on deep generative modeling tasks using various benchmark datasets such as CIFAR10, CelebA, and Tiny ImageNet.
- Their work provides an efficient way to estimate SW distances in high-dimensional settings that can benefit many applications such as deep learning models that use mini-batch approaches.

## Summary

The Hierarchical Sliced Wasserstein (HSW) distance is a new metric between measures proposed in a recent paper by Khai Nguyen, Tongzheng Ren, Huy Nguyen, Litu Rout, Tan Nguyen, and Nhat Ho. The sliced Wasserstein (SW) distance has been widely used in various application scenarios due to its ability to scale to a large number of supports without suffering from the curse of dimensionality. However, estimating the sliced Wasserstein requires a relatively large number of projections in high-dimensional settings. This becomes the main computational bottleneck for applications where the number of supports is relatively small compared with the dimension. To address this issue, the authors propose deriving projections by linearly and randomly combining a smaller number of projections called bottleneck projections. They explain how these projections are used by introducing Hierarchical Radon Transform (HRT), which is constructed by applying Radon Transform variants recursively. The approach is formulated into the new metric between measures named HSW distance. By proving the injectivity of HRT, they derive the metricity of HSW and investigate its theoretical properties including its connection to SW variants and its computational and sample complexities. Finally, they compare the computational cost and generative quality of HSW with conventional SW on deep generative modeling tasks using various benchmark datasets such as CIFAR10, CelebA, and Tiny ImageNet. The authors' proposed method addresses the limitations of conventional SW distance estimation in high-dimensional settings where computing transportation costs between one-dimensional representations can be computationally expensive. By using bottleneck projections derived from linear combinations of fewer Radon Transform projections through Hierarchical Radon Transform (HRT), they achieve comparable results with conventional SW while reducing computation time significantly. Overall, their work provides an efficient way to estimate SW distances in high-dimensional settings that can benefit many applications such as deep learning models that use mini-batch approaches.